Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(27,36)(29,39)(31,32)(34,40);(1,3)(2,4)(5,8)(6,7)(11,17)(15,21)(18,23)(22,24)(25,26,30,35)(27,32,36,31)(28,37,38,33)(29,34,39,40);(1,2)(3,4)(5,7)(6,8)(10,14)(15,21)(16,19)(22,24)(25,27,30,36)(26,31,35,32)(28,34,38,40)(29,33,39,37);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,35)(37,39)(38,40);(1,4,5,6)(2,3,7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,38)(26,33,35,37)(27,34,36,40)(29,31,39,32);(1,3)(2,4)(5,8)(6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29)(26,34)(27,33)(28,32)(30,39)(31,38)(35,40)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9634163);